báo cáo hóa học:" Research Article A General Iterative Approach to Variational Inequality Problems and Optimization Problems Jong Soo Jung" ppt

Volume 2011, Article ID 284363, 20 pagesdoi:10.1155/2011/284363 Research Article A General Iterative Approach to Variational Inequality Problems and Optimization Problems Jong Soo Jung D

Volume 2011, Article ID 284363, 20 pages

doi:10.1155/2011/284363

Research Article

A General Iterative Approach to Variational

Inequality Problems and Optimization Problems

Jong Soo Jung

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Correspondence should be addressed to Jong Soo Jung,jungjs@mail.donga.ac.kr

Received 4 October 2010; Accepted 14 November 2010

Academic Editor: Jen Chih Yao

Copyrightq 2011 Jong Soo Jung This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We introduce a new general iterative scheme for finding a common element of the set of solutions

of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem Applications of the main result are also given

1 Introduction

Let H be a real Hilbert space with inner product ·, · and induced norm  ·  Let C be a nonempty closed convex subset of H and S : C → C be self-mapping on C We denote by

F S the set of fixed points of S and by P C the metric projection of H onto C.

Let A be a nonlinear mapping of C into H The variational inequality problem is to find a u ∈ C such that

We denote the set of solutions of the variational inequality problem1.1 by VIC, A The

variational inequality problem has been extensively studied in the literature; see1 5 and the references therein

Recently, in order to study the problem1.1 coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem1.1 and the set of fixed points of nonexpansive mappings; see 6 9 and the references therein In particular, in 2005, Iiduka and Takahashi 8

introduced an iterative scheme for finding a common point of the set of fixed points of a

nonexapansive mapping S and the set of solutions of the problem1.1 for an inverse-strong

monotone mapping A: x1∈ C and

x n1 α n x  1 − α n SP C x n − λ n Ax n , n ≥ 1, 1.2

where{α n } ⊂ 0, 1 and {λ n } ⊂ 0, 2α They proved that the sequence generated by 1.2

strongly converges strongly to P F S∩VIC,A x In 2010, Jung10 provided the following new composite iterative scheme for the fixed point problem and the problem1.1: x1 x ∈ C and

y n  α n f x n   1 − α n SP C x n − λ n Ax n ,

x n11− β n



y n  β n SP C

y n − λ n Ay n

where f is a contraction with constant k ∈ 0, 1,{α n },{β n } ∈ 0, 1, and {λ n } ⊂ 0, 2α He

proved that the sequence{x n} generated by 1.3 strongly converges strongly to a point in

F S ∩ VIC, A, which is the unique solution of a certain variational inequality.

On the other hand, the following optimization problem has been studied extensively

by many authors:

min

x∈Ω

μ

2Bx, x 1

where Ω  ∞

n1C n , C1, C2, are infinitely many closed convex subsets of H such that

∞

n1C n /  ∅, u ∈ H, μ ≥ 0 is a real number, B is a strongly positive bounded linear operator on

H i.e., there is a constant γ > 0 such that Bx, x ≥ γx2, for all x ∈ H, and h is a potential function for γf i.e., h x  γfx for all x ∈ H For this kind of optimization problems,

see, for example, Deutsch and Yamada11, Jung 10, and Xu 12,13 when Ω N

i1C iand

h x  x, b for a given point b in H.

In 2007, related to a certain optimization problem, Marino and Xu14 introduced the following general iterative scheme for the fixed point problem of a nonexpansive mapping:

x n1 α n γf x n   I − α n B Sx n , n ≥ 0, 1.5

where {α n } ∈ 0, 1 and γ > 0 They proved that the sequence {x n} generated by 1.5 converges strongly to the unique solution of the variational inequality



B − γfx∗, x − x∗

which is the optimality condition for the minimization problem

min

1

where h is a potential function for γf The result improved the corresponding results of

Moudafi15 and Xu 16

In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem1.1 for an inverse-strongly monotone mapping and the set

of fixed points of a nonexapansive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem Applications of the main result are also discussed Our results improve and complement the corresponding results of Chen et al.6, Iiduka and Takahashi8, Jung 10, and others

2 Preliminaries and Lemmas

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H We write

x n  x to indicate that the sequence {x n } converges weakly to x x n → x implies that {x n}

converges strongly to x.

First we recall that a mapping f : C → C is a contraction on C if there exists a constant

k ∈ 0, 1 such that fx − fy ≤ kx − y, x, y ∈ C A mapping T : C → C is called

nonexpansive if Tx − Ty ≤ x − y, x, y ∈ C We denote by FT the set of fixed points of T For every point x ∈ H, there exists a unique nearest point in C, denoted by P C x, such

that

for all y ∈ C P C is called the metric projection of H onto C It is well known that P C is

nonexpansive and P C satisfies



x − y, P C x − P C



y

≥P C x − P C

for every x, y ∈ H Moreover, P C x is characterized by the properties:

x − y2

≥ x − P C x2y − P C x2,

In the context of the variational inequality problem for a nonlinear mapping A, this implies

that

u ∈ VIC, A ⇐⇒ u  P C u − λAu, for any λ > 0. 2.4

It is also well known that H satisfies the Opial condition, that is, for any sequence {x n} with

x n  x, the inequality

lim inf

n→ ∞ x n − x < lim inf n→ ∞ x n − y 2.5

holds for every y ∈ H with y / x.

A mapping A of C into H is called inverse-strongly monotone if there exists a positive real number α such that



for all x, y ∈ C; see 4,7,17 For such a case, A is called α-inverse-strongly monotone We know that if A  I ưT, where T is a nonexpansive mapping of C into itself and I is the identity mapping of H, then A is 1/2-inverse-strongly monotone and VIC, A  FT A mapping A

of C into H is called strongly monotone if there exists a positive real number η such that



for all x, y ∈ C In such a case, we say A is η-strongly monotone If A is η-strongly monotone and κ-Lipschitz continuous, that is, Ax ư Ay ≤ κx ư y for all x, y ∈ C, then A is η/κ2

-inverse-strongly monotone If A is an α inverse-strongly monotone mapping of C into H, then it is obvious that A is 1/α-Lipschitz continuous We also have that for all x, y ∈ C and

λ > 0,

I ư λAx ư I ư λAy2x ư y

ư λAx ư Ay2

x ư y2ư 2λx ư y, Ax ư Ay  λ2Ax ư Ay2

≤x ư y2

 λλ ư 2αAx ư Ay2

.

2.8

So, if λ ≤ 2α, then I ư λA is a nonexpansive mapping of C into H The following result for the

existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda9

Proposition 2.1 Let C be a bounded closed convex subset of a real Hilbert space and let A be an

α-inverse-strongly monotone mapping of C into H Then, VI C, A is nonempty.

A set-valued mapping T : H → 2H is called monotone if for all x, y ∈ H, f ∈ Tx, and

g ∈ Ty imply xưy, fưg ≥ 0 A monotone mapping T : H → 2 H is maximal if the graph GT

of T is not properly contained in the graph of any other monotone mapping It is known that

a monotone mapping T is maximal if and only if for x, f ∈ H ×H, xưy, f ưg ≥ 0 for every

y, g ∈ GT implies f ∈ Tx Let A be an inverse-strongly monotone mapping of C into H and let N C v be the normal cone to C at v, that is, N C v  {w ∈ H : vưu, w ≥ 0, for all u ∈ C},

and define

Tv

⎧

⎨

⎩

Av  N C v, v ∈ C,

Then T is maximal monotone and 0 ∈ Tv if and only if v ∈ VIC, A; see 18,19

We need the following lemmas for the proof of our main results.

Lemma 2.2 In a real Hilbert space H, there holds the following inequality:

x  y2

for all x, y ∈ H.

Lemma 2.3 Xu 12 Let {s n } be a sequence of nonnegative real numbers satisfying

where {λ n } and {β n } satisfy the following conditions:

i {λ n } ⊂ 0, 1 and∞

n1λ n  ∞ or, equivalently,∞

n11 − λ n   0;

ii lim supn→ ∞β n /λ n  ≤ 0 or∞

n1|β n | < ∞;

iii γ n ≥ 0 n ≥ 1, ∞

n1γ n < ∞.

Then lim n→ ∞s n  0.

Lemma 2.4 Marino and Xu 14 Assume that A is a strongly positive linear bounded operator on

a Hilbert space H with constant γ > 0 and 0 < ρ ≤ B−1 Then I − ρB ≤ 1 − ργ.

The following lemma can be found in20,21 see alsoLemma 2.2in22

Lemma 2.5 Let C be a nonempty closed convex subset of a real Hilbert space H, and let g : C →

R ∪ {∞} be a proper lower semicontinunous differentiable convex function If x∗is a solution to the minimization problem

g x∗  inf

then



g x, x − x∗

In particular, if x∗solves the optimization problem

min

x ∈C

μ

2Bx, x 1

then



uγf−I  μBx∗, x − x∗

where h is a potential function for γf.

3 Main Results

In this section, we present a new general composite iterative scheme for inverse-strongly monotone mappings and a nonexpansive mapping

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H such that C ± C ⊂

C Let A be an α-inverse-strongly monotone mapping of C into H and S a nonexpansive mapping of

C into itself such that F S ∩ VIC, A / ∅ Let u ∈ C and let B be a strongly positive bounded linear

operator on C with constant γ ∈ 0, 1 and f a contraction of C into itself with constant k ∈ 0, 1.

Assume that μ > 0 and 0 < γ < 1  μγ/k Let {x n } be a sequence generated by

x1 x ∈ C,

y n  α n



u  γfx nI − α n



I  μBSP C x n − λ n Ax n ,

x n11− β n



y n  β n SP C

y n − λ n Ay n

, n ≥ 1,

IS

where {λ n } ⊂ 0, 2α, {α n } ⊂ 0, 1, and {β n } ⊂ 0, 1 Let {α n }, {λ n }, and {β n } satisfy the following

conditions:

i α n → 0 n → ∞;∞n1α n  ∞;

ii β n ⊂ 0, a for all n ≥ 0 and for some a ∈ 0, 1;

iii λ n ∈ c, d for some c, d with 0 < c < d < 2α;

iv∞n1|α n1− α n | < ∞,∞n1|β n1− β n | < ∞,∞n1|λ n1− λ n | < ∞.

Then {x n } converges strongly to q ∈ FS ∩ VIC, A, which is a solution of the optimization problem

min

μ

2Bx, x 1

where h is a potential function for γf.

Proof We note that from the control conditioni, we may assume, without loss of generality,

that α n ≤ 1  μB−1 Recall that if B is bounded linear self-adjoint operator on H, then

Observe that



I − α n



I  μBu, u

 1 − α n − α n μ Bu, u

≥ 1 − α n − α n μ B

≥ 0,

3.2

which is to say that I − α n I  μB is positive It follows that

I − α n



I  μBu, u

: u ∈ H, u  1

 sup 1− α n − α n μ Bu, u : u ∈ H, u  1

≤ 1 − α n



1 μγ

< 1 − α n



1 μγ.

3.3

Now we divide the proof into several steps

Step 1 We show that {x n } is bounded To this end, let z n  P C x n − λ n Ax n  and w n  P C y n−

λ n Ay n  for every n ≥ 1 Let p ∈ FS ∩ VIC, A Since I − λ n A is nonexpansive and p 

P C p − λ n Ap from 2.4, we have

z n − p ≤ x n − λ n Ax n −p − λ n Ap

Similarly, we have

Now, set B  I  μB Let p ∈ FS ∩ VIC, A Then, from IS and 3.4, we obtain

y n − p α n u  α n



γf x n  − BpI − α n B

Sz n − p

≤1−1 μγα nz n − p  α n u

 α n γf x n  − f

p   α nγf

p

− Bp

≤1−1 μγα nz n − p  α n u

 α n γkx n − p  α nγf

p

− Bp

1−1 μγ − γkα nx n1− p

1 μγ − γkα n



γfp

− Bp  u



1 μγ − γk .

3.6

From3.5 and 3.6, it follows that

x n1− p 1− β n

y n − p β n



Sw n − p

≤1− β ny n − p  β nw n − p

≤1− β ny n − p  β ny n − p

y n − p

≤ max

⎧

⎨

⎩x n − p,γfp

− Bp  u



1 μγ − γk

⎫

⎬

⎭.

3.7

By induction, it follows from3.7 that

x n − p ≤ max⎧⎨⎩x1− p,γfp

− Bp  u



1 μγ − γk

⎫

⎬

Therefore, {x n } is bounded So {y n }, {z n }, {w n }, {fx n }, {Ax n }, {Ay n }, and {BSz n} are bounded Moreover, sinceSz n − p ≤ x n − p and Sw n − p ≤ y n − p, {Sz n } and {Sw n} are also bounded And by the conditioni, we have

y n − Sz n   α nu  γfx n

−I  μBSz n

 α n

u  γfx n− BSz n −→ 0 as n −→ ∞. 3.9

Step 2 We show that lim n→ ∞x n1−x n  0 and limn→ ∞y n1−y n   0 Indeed, since I −λ n A

and P C are nonexpansive and z n  P C x n − λ n Ax n, we have

z n − z n−1 ≤ x n − λ n Ax n  − x n−1− λ n−1Ax n−1

≤ x n − x n−1  |λ n − λ n−1|Ax n−1. 3.10

Similarly, we get

w n − w n−1 ≤y n − y n−1  |λ n − λ n−1|Ay n−1. 3.11 Simple calculations show that

y n − y n−1 α n



u  γfx nI − α n B

Sz n − α n−1

u  γfx n−1−I − α n−1B

Sz n−1

 α n − α n−1u  γfx n−1 − BSz n−1



 α n γ

f x n  − fx n−1

I − α n B

Sz n − Sz n−1.

3.12

So, we obtain

y n − y n−1 ≤ |α n − α n−1|u  γf x n−1 BSz

n−1

 α n γk x n − x n−1 1−1 μγα n

z n − z n−1

≤ |α n − α n−1|u  γf x n−1 BSz

n−1

 α n γk x n − x n−1 1−1 μγα n

x n − x n−1

 |λ n − λ n−1|Ax n−1.

3.13

Also observe that

x n1− x n1− β n



y n − y n−1

β n − β n−1

Sw n−1− y n−1

By3.11, 3.13, and 3.14, we have

x n1− x n ≤1− β ny n − y n−1  β n − β n−1Sx n−1 y n−1

 β n w n − w n−1

≤1− β ny n − y n−1  β ny n − y n−1  β n |λ n − λ n−1|Ay n−1

β n − β n−1Sw n−1 y n−1

≤y n − y n−1  |λ n − λ n−1|Ay n−1  β n − β n−1Sw n−1 y n−1

≤1−1 μγ − γkα n



x n − x n−1

 |α n − α n−1|u  γf x n−1 BSz

n−1

 |λ n − λ n−1|Ay n−1  Ax n−1β n − β n−1Sw n−1 y n−1

≤1−1 μγ − γkα n



x n − x n−1

 M1|α n − α n−1|  M2|λ n − λ n−1|  M3β n − β n−1,

3.15

where M1 sup{u  γfx n   BT n z n  : n ≥ 1}, M2 sup{Ay n   Ax n  : n ≥ 1}, and

M3 sup{Sw n   y n  : n ≥ 1} From the conditions i and iv, it is easy to see that

lim

n→ ∞



1 μγ − γkα n  0, ∞

n1



1 μγ − γkα n  ∞,

∞



n2



M1|α n − α n−1|  M2|λ n − λ n−1|  M3β n − β n−1< ∞. 3.16

ApplyingLemma 2.3to3.15, we obtain

lim

Moreover, by3.10 and 3.13, we also have

lim

n→ ∞z n1− z n   0, lim

n→ ∞y n1− y n   0. 3.18

Step 3 We show that lim n→ ∞x n − y n  0 and limn→ ∞x n − Sz n  0 Indeed,

x n1− y n   β nSw n − y n

≤ β n



Sw n − Sz n Sz n − y n

≤ aw n − z n Sz n − y n

≤ ay n − x n   Sz n − y n

≤ ay n − x n1  x n1− x n Sz n − y n

3.19

which implies that

x n1− y n ≤ a

1− a



x n1− x n Sz n − y n. 3.20

Obviously, by3.9 andStep 2, we havex n1− y n  → 0 as n → ∞ This implies that

x n − y n  ≤ x n − x n1 x n1− y n  −→ 0 as n −→ ∞. 3.21

By3.9 and 3.21, we also have

x n − Sz n ≤x n − y n   y n − Sz n  −→ 0 as n −→ ∞. 3.22